On the Theory of the Width of Atomic Spectral Lines

I. I. Sobel’man

Introduction

The present article is devoted to the theory of the broadening of spectral lines caused by the interaction of a radiating atom with the particles surrounding it—other atoms and molecules, ions, and electrons. The main attention is given to discussion of the new results and advances achieved in the theory of this phenomenon after the publication of the widely known reviews by Weisskopf \(^{1}\) (1933) and Margenau and Watson \(^{4}\) (1936). In the exposition use is made of the review article by Unsöld \(^{2}\) (1943), published somewhat later, especially in the part illustrating the theory by experiment. The discussion of experimental work is carried out only in connection with the theory and does not claim to be complete.

At first the theory was developed on the basis of an analogy between an atom and a classical oscillator. The distribution of intensity in the emission line of an oscillator, whose oscillation in the general case may be written in the form:

\[ f(t)=A(t)\exp\left[i\int_{-\infty}^{t}\omega_p(t')\,dt'\right] \tag{1} \]

is determined, as is known, by expanding \(f(t)\) in a Fourier integral

\[ f_{\omega}=\lim_{T\to\infty}\frac{1}{(2\pi T)^{1/2}}\int_{-T/2}^{T/2} f(t)e^{-i\omega t}\,dt, \tag{2} \]

\[ I(\omega)\,d\omega \sim \overline{f_{\omega}f_{\omega}^{*}}\,d\omega. \tag{3} \]

If the atom is replaced by an equivalent atomic oscillator with amplitude \(A(t)\) and instantaneous frequency \(\omega_p(t)\), equal to the difference

of the initial and final terms, then the calculation of \(I(\omega)\,d\omega\) reduces to relations (2) and (3). To determine \(A(t)\) and \(\omega_p(t)\), it is necessary to make certain assumptions about the character of the interaction. In addressing this question, two completely different concepts have taken shape: the impact concept (Lorentz, Lenz and Weisskopf\(^{1,2}\)) and the statistical concept (Holtsmark, Kuhn and Margenau\(^{1,2,4}\)).

In the first, it is assumed that the decisive role is played by the perturbation of the oscillator phase at the instant of collision. Between two collisions, which are regarded as instantaneous, the perturbation may be neglected and one may set \(A=\mathrm{const}\) and \(\omega_p=\omega_0\), where \(\omega_0\) is the unperturbed value of the oscillator frequency. Owing to collisions, the oscillation of the oscillator is split into a series of noninterfering (incoherent) trains, which causes the broadening of the line.

The second concept is based on the assumption that the atom is continually subject to an external action, as a result of which \(\omega_p\) changes continuously in time. Assuming this change to be slow, one may regard \(I(\omega)\,d\omega\) as simply proportional to the probability \(W(\omega)\,d\omega\) with which \(\omega_p\) assumes values contained in the interval \(\omega,\omega+d\omega\).

As was to be expected, the two concepts give sharply different expressions for the shape and width of the line.

Analysis of the experimental facts has shown that both the impact and the statistical theory are capable of satisfactorily explaining a number of experimental facts (see, in this connection,\(^{1}\) and \(^{4}\)). At the same time, questions requiring clarification have emerged. Thus, the existence of two diametrically opposed concepts raises the question of the limits of their applicability, the relation between them, and their connection with the general relations (2) and (3). For a long time it was not possible to interpret the experimentally observed line shift at low pressures, proportional to the density of the perturbing particles \(N\).

In the time that has elapsed since the publication of the reviews\(^{1}\) and \(^{4}\), papers by Lindholm\(^{13}\), Foley\(^{30}\), and Anderson\(^{12}\) have appeared, clarifying the question of the line shift. In the same way, it has been possible to bring clarity to the question of the relation between the impact and statistical theories\(^{2,39}\).

A special place is occupied by the problem of constructing a consistent quantum-mechanical theory. The first step in this direction was taken by Weisskopf\(^{3}\). He showed that the loss of monochromaticity is connected with the transfer of excitation energy to external degrees of freedom; moreover, the results of the quantum-mechanical treatment of this effect in the case when the relative motion of the colliding particles is quasiclassical are completely equivalent to relations (2) and (3) for a classical oscillator.

This question was subsequently considered repeatedly from various points of view \(^{12, 30, 37, 48}\).

Common to all these works is the assumption of the quasi-classical nature of the relative motion of the atom and the perturbing particle. In all cases this assumption makes it possible to reduce the calculation of \(I(\omega)\,d\omega\) to a Fourier analysis of the oscillations of an equivalent oscillator.

Considerable difficulties arise when the quasi-classical approximation is inapplicable. This occurs in interaction with light particles, especially with electrons. At present this question has not yet received a satisfactory solution, although it is possible to outline a scheme of solution that gives grounds for hoping for success.

Similarly, a number of problems connected with the behavior of an atom in an inhomogeneous field varying with time still remain unresolved. Here, however, some progress has recently been made \(^{32—34, 26, 46, 20}\).

I. COLLISION THEORY

The collision theory (in the form in which it was formulated by Lorentz and then by Lenz and Weisskopf \(^{1}\)) is based on the assumption that the decisive factor in the broadening of a line is the disruption of the coherence of the oscillations of the atomic oscillator during a collision.

Lorentz did not specify the mechanism of the collision. It was simply assumed that, as a result of collisions, the oscillator oscillation \(e^{i\omega_0 t}\) is broken up into a series of independent trains with mean duration \(\tau\). The representation of such a set of trains by a Fourier integral gives an intensity distribution in the line, symmetric with respect to \(\omega_0\),

\[ I(\omega)\,d\omega=\frac{d\omega}{\pi \tau \{(\omega-\omega_0)^2+(1/\tau)^2\}} \tag{4} \]

with width \(\gamma=2/\tau\). \(1/\tau\) is the number of collisions in 1 sec, equal to \(N v \sigma\), where \(v\) is the velocity of the relative motion of the colliding particles, and \(\sigma\) is the effective collision cross section. The Lorentz theory did not give the magnitude \(\sigma\). Naturally, the question arose: how should one approach the estimation of \(\sigma\)?

It is obvious that there is no basis for equating \(\sigma\) to its gas-kinetic value, since, according to the basic assumption of the collision theory, a collision should be understood as an encounter in which the coherence of the oscillations of the atomic oscillator is destroyed. Likewise, \(\sigma\) cannot be equated to the cross section for quenching resonance fluorescence, since experiment shows that this process is not decisive: thus, the addition of He and \(N_2\) to Na broadens the lines to the same extent, whereas

the efficiency of quenching of resonance fluorescence in them is sharply different.

Lenz and Weisskopf solve this question in the following way. When the perturbing particle flies past, the frequency of the atomic oscillator is shifted, as a result of which the phase of the oscillator acquires an additional increment. If this additional phase shift \(\eta\) is sufficiently large, i.e., exceeds some value \(\eta_0\), then the coherence of the oscillations is destroyed. Thus, flights with \(\eta \gg \eta_0\) must be regarded as collisions.

Putting the frequency shift \(\chi = C_n/R^n\), we calculate \(\eta\) for a flyby at impact distance \(\rho\)

\[ \eta(\rho)=\int_{-\infty}^{\infty}\frac{C_n dt}{\{\rho^2+v^2t^2\}^{n/2}} = a_n \frac{C_n}{v \rho^{n-1}}, \tag{5} \]

where

\[ a_n=\sqrt{\pi}\,\frac{\Gamma\left(\frac{n-1}{2}\right)}{\Gamma(n/2)}. \tag{6} \]

According to Weisskopf, one should put \(\eta_0=1\). This makes it possible to determine the largest value of \(\rho\) for which flybys are still effective (the so-called Weisskopf radius \(\rho_0\)).

\[ \rho_0=\left(\frac{a_n C_n}{v}\right)^{1/(n-1)}. \tag{7} \]

Putting \(\sigma=\pi \rho_0^2\), we obtain for the line width the expression

\[ \gamma=2\pi Nv\left(\frac{a_n C_n}{v}\right)^{\frac{2}{n-1}}. \tag{8} \]

For the cases \(n=2, 3, 4, 5\), and \(6\), which may be of interest, \(a_n\) is equal to \(\pi\), \(2\), \(\pi/2\), \(4/3\), and \(3\pi/8\), respectively.*

The conception set forth above suffers from the drawback that flybys outside \(\rho_0\) are completely excluded from consideration. In addition, because of the arbitrariness in the choice of the limiting value \(\eta_0\) (why, for example, one, and not \(\pi\) or \(\pi/2\)?) relation (8) can give no more than the order of magnitude of \(\gamma\).

* It must be emphasized that the indicated definition of \(\sigma\) is not the only possible one; we shall become acquainted with another approach to this same problem below, in § 4 of Section IV.

An extremely simple and elegant solution of the question of impact broadening of intensity, free from these shortcomings, can be obtained by means of correlation theory ^{30,12}. (Still earlier, a similar consideration was carried out in ^{13}.)

Let us write the oscillation of the atomic oscillator in the form:

\[ f(t)=e^{i\omega_0 t+i\eta(t)}, \tag{9} \]

where \(\eta(t)\) is the total phase shift caused by the perturbation during the time preceding \(t\).

Using the known relation between the spectral density of a process and its correlation function*), we obtain:

\[ I(\omega)=\int_{-\infty}^{\infty}\exp\{i(\omega_0-\omega)s\}\,\varphi(s)\,ds, \tag{10} \]

where

\[ \varphi(s)=\int_{-\infty}^{\infty}\exp\{i[\eta(t)-\eta(t+s)]\}\,dt. \tag{11} \]

Denote \(\eta(t+s)-\eta(t)\), i.e. the total phase shift over the time interval \(t, t+s\), by \(\eta(t,s)\). Then the integrand in (11) can be written in the form \(e^{-i\eta(t,s)}\).

Form the difference

\[ e^{-i\eta(t,s+ds)}-e^{-i\eta(t,s)} = e^{-i\eta(t,s)}\{e^{-i\eta'}-1\}. \]

Here \(\eta'\) is the additional phase shift during the time \(ds\). Averaging this expression over time, in accordance with (11), we obtain:

\[ \varphi(s+ds)-\varphi(s)=d\varphi(s) = \overline{e^{-i\eta(t,s)}\{e^{-i\eta'}-1\}}. \tag{12} \]

In performing the averaging in (12), one has to make use of the assumptions of impact theory. If the phase during a collision changes

*) The correlation function of a process \(f(t)\) is the function

\[ \varphi(s)=\int_{-\infty}^{\infty} f(t)f^*(t+s)\,dt, \]

and the Fourier component of \(\varphi(s)\) determines the spectral density of the process:

\[ \varphi_\omega=\int_{-\infty}^{\infty} e^{-i\omega s}\varphi(s)\,ds=I(\omega). \]

Compare, for example, ^{21}.

instantaneously, then \(\eta'\) does not depend on the value of the phase at the instant \(t-s\), and the averaging of the two factors may be carried out separately. By definition \(\overline{e^{-i\eta'(t,s)}}=\varphi(s)\). In calculating the second factor, we replace averaging over time by averaging over all possible passages. The number of passages through an annular element \(2\pi\rho\,d\rho\) during the time \(ds\) will be equal to \(2\pi\rho\,d\rho\cdot Nv\,ds\). Therefore

\[ \overline{\{e^{-i\eta'}-1\}} = Nv\,ds\int_0^\infty \{e^{-i\eta(\rho)}-1\}\,2\pi\rho\,d\rho = -Nv\,ds\,\{\sigma_r+i\sigma_i\}, \]

where

\[ \sigma_r=2\pi\int_0^\infty \{1-\cos\eta(\rho)\}\rho\,d\rho, \tag{13} \]

\[ \sigma_i=2\pi\int_0^\infty \sin\eta(\rho)\cdot\rho\,d\rho. \tag{14} \]

It is now easy to obtain:

\[ \varphi(s)=\exp\{-Nv(\sigma_r+i\sigma_i)s\}, \tag{15} \]

after which from (10) there follows directly

\[ I(\omega)\,d\omega= \frac{Nv\sigma_r/\pi}{(\omega-\omega_0-Nv\sigma_i)^2+(Nv\sigma_2)^2}\,d\omega. \tag{16} \]

This expression is analogous to (4), but now \(\gamma=2Nv\sigma_r\), and the maximum of the line is shifted from \(\omega_0\) by the amount \(\Delta=Nv\sigma_i\).

Let us estimate the contribution of distant and close passages to \(\sigma_r\) and \(\sigma_i\). In Fig. 1 an approximate form of the integrands in (13) and (14) is given for the case \(n=4\).

For \(0\le \rho\le \rho_0\), \(\eta(\rho)\ge 1\), consequently \(\cos\eta(\rho)\) and \(\sin\eta(\rho)\) oscillate rapidly, in accordance with which

\[ \int_0^{\rho_0}\{1-\cos\eta(\rho)\}\,2\pi\rho\,d\rho\simeq \pi\rho_0^2, \tag{17} \]

\[ \int_0^{\rho_0}\sin\eta(\rho)\cdot 2\pi\rho\,d\rho\simeq 0, \]

i.e. in complete agreement with Weisskopf’s theory, passages inside \(\rho_0\) give a line broadening \(\simeq 2\pi\rho_0^2Nv\) and give no shift. Conversely, passages outside \(\rho_0\) \((\eta(\rho)<1)\) are practically insignificant for the broadening, since

\[ \int_{\rho_0}^{\infty}\{1-\cos\eta(\rho)\}\,2\pi\rho\,d\rho \ll \pi\rho_0^2, \]

but give the principal contribution to \(\sigma_i\). It is easy to clarify the physical meaning of the line shift. Suppose that we are dealing with passages in which

ON THE THEORY OF THE WIDTH OF ATOMIC SPECTRAL LINES

for which \(\eta \ll 1\). In this case (14) gives

\[ \sigma_i = 2\pi \int_0^\infty \eta(\rho)\cdot \rho\, d\rho = \bar{\eta}, \]

where \(\bar{\eta}\) is the mean phase shift produced by the perturbation per unit time. Thus, an additional phase increment with mean rate \(\bar{\eta}\) is equivalent to a change in the oscillator frequency by the amount \(\bar{\eta}\). This result can be obtained directly from (9), if \(\eta(t)\) is approximately replaced by \(\bar{\eta}\cdot t\). (The calculation of \(\Delta\) was carried out in an analogous way in the work \({}^{11}\).) It is seen from (14) that \(\sigma_i\) changes sign when the sign of \(\eta\) changes. If collisions of different types, accompanied by phase shifts \(\eta>0\) and \(\eta<0\), are equally probable, then the total shift of the line is equal to zero.

Let us also note that for large values of \(\eta(\rho)\), owing to the strong oscillation of the factor \(\{1-\cos\eta(\rho)\}\), \(\sigma_r\) is only weakly sensitive to changes in the form of the function \(\eta(\rho)\). In other words, knowledge of the exact dependence \(\eta(\rho)\) in this region is immaterial.

Fig. 1.

We shall give the values of the width \(\gamma\) and of the shift of the line maximum \(\Delta\) for a number of concrete cases:

\[ \left. \begin{array}{ll} n=2 & \gamma_2 = 2\pi^3 N C_2^2/v \quad *),\\ n=3 & \gamma_3 = 2\pi^2 C_3 N,\\ n=4 & \gamma_4 = 11.4\, C_4^{2/3} v^{1/3} N;\quad \Delta_4 = 9.8\, C_4^{2/3} v^{1/3} N;\quad \gamma/\Delta = 1.16,\\ n=6 & \gamma_6 = 8.16\, C_6^{2/5} v^{3/5} N;\quad \Delta_6 = 2.96\, C_6^{2/5} v^{3/5} N;\quad \gamma/\Delta = 2.8. \end{array} \right\} \tag{18} \]

*) For \(n=2\) the integral (13) diverges. This divergence has no physical meaning, since at large \(\rho\) (of the order of the mean distance to the perturbing particles) the impact theory is already inapplicable. Since the introduction of a finite radius of action of the forces, which makes it possible to remove this divergence, introduces an element of uncertainty, (15) gives the value \(\gamma_2\), obtained from (8). The calculation of \(\Delta\) for \(n=2\) and 3 is of no practical interest.

The obtained values of \(\gamma_3\), \(\gamma_4\), and \(\gamma_6\) are somewhat larger than those of Weisskopf—respectively by \(\pi/2\), \(1.35\), and \(1.2\) times. In order to bring these values into agreement with Weisskopf’s formula (8), when determining \(\rho_0\) one must set \(\eta_0=0.64\) (for \(n=3,4\)) and \(\eta_0=0.61\) (for \(n=6\)) instead of unity. The linear dependence of \(\gamma\) and \(\Delta\) on \(N\), as well as the relation between \(\gamma\) and \(\Delta\), agree well with the experimental data \(^{2,4,18\text{--}20}\).

The expression (16) obtained for \(I(\omega)\,d\omega\) coincides exactly with the Lorentz formula (4), if in the latter one substitutes \(\sigma=\sigma_r+i\sigma_i\). Thus the effective collision cross section turns out to be complex (\(\sigma_r\) and \(\sigma_i\), of course, being real). Below, in Section V, the connection between the classical theory developed in this section and the general theory of collisions will be established. In particular, the relation between \(\sigma_r\) and \(\sigma_i\) and the effective scattering cross section will be clarified.

II. STATISTICAL THEORY

The problem of the influence of pressure can also be approached from another point of view. The radiating atom is in an external field. The character of this field depends on the type of particles surrounding the atom, on their density, and on their velocities of motion. The presence of the field leads to a displacement of the terms and, consequently, to a shift of the frequency \(\omega_p\) of the atomic oscillator. If the external field is quasistatic, i.e. changes sufficiently slowly (the time of appreciable change of the field, or, in other words, of change in the arrangement of the surrounding particles, is large compared with \(1/\omega_0\)), then one may assume that \(I(\omega)\,d\omega\) is simply proportional to the statistical weight of the configuration of perturbing particles for which \(\omega_p\) lies in the interval \(\omega,\omega+d\omega\). Let us consider first of all the influence of the nearest particle \(^{2}\). The probability that the nearest particle is at a distance \(R,R+dR\) from the atom, \(W(R)\,dR\), is equal to \(4\pi R^2 N\,dR\,e^{-4\pi R^3N/3}\).

If we introduce the notation \(\omega-\omega_0=C_n/R^n\) and \(\Delta\omega=C_n/\overline{R}^{\,n}\), where

\[ \overline{R}=(3/4\pi N)^{1/3} \]

is the mean value of \(R\) at the given density, then it is easy to obtain

\[ I(\omega)\,d\omega= \frac{4\pi N C_n^{3/n}} {n(\omega-\omega_0)^{3/n+1}} e^{-\left(\frac{\Delta\omega}{\omega-\omega_0}\right)^{3/n}} \,d\omega . \tag{19} \]

It is evident that the nearest particle produces the largest shifts, and consequently (19) is valid for the wings of the line \(R\ll \overline{R}\) and \((\omega-\omega_0)\gg \Delta\omega\). Therefore for the wings of the line one may also use the expression

\[ I(\omega)\,d\omega= \frac{4\pi N C_n^{3/n}} {n(\omega-\omega_0)^{3/n+1}}\,d\omega . \tag{20} \]

The internal parts of a line are produced by the combined action of many particles. In computing \(I(\omega)\,d\omega\), it is first necessary to determine how the superposition of perturbations takes place. The influence of electric fields (the linear and quadratic Stark effects, \(n=2\) and \(n=4\)) was considered by Holtsmark\(^2\). Later Margenau\(^5\) solved the analogous problem for the van der Waals interaction \(n=6\), and also for the cases \(n=3\) and \(n=5\). In the last two cases additivity of the interaction and equal probability of positive and negative shifts of the oscillator frequency were assumed. We shall not dwell on the exposition of these works, since all these calculations are of no practical interest. As will be shown in the next section, at low pressures the statistical theory determines the distribution of intensity in the wing of the line, the latter being produced by the nearest particle, which makes it possible to use the simple expression (20). This situation persists as long as \(N\) is sufficiently small, namely, \(N \ll 1/\rho_0^3\). Beginning with \(N \simeq 1/\rho_0^3\), and for \(N > 1/\rho_0^3\), the mean distance between particles becomes of the order of atomic dimensions. Under these conditions the assumptions underlying the calculations (a simple interaction law \(1/R^n\), homogeneity of the external field, etc.; see more detail in Section IV) cease to be fulfilled.

III. THE RELATION AND LIMITS OF APPLICABILITY OF THE IMPACT AND STATISTICAL THEORIES. THE ROLE OF THE DOPPLER EFFECT IN LINE BROADENING

The question of the limits of applicability of the impact and statistical theories has been discussed more than once. Thus, in works\(^ {30,40}\) criteria were proposed that determine the transition from impact broadening to statistical broadening as \(N\) increases, and the question of the applicability of one theory or the other was decided at once for the whole line. On the other hand, Unzöld\(^2\), on the basis of works\(^ {7,31,32,33,34,11}\), showed that in the wing of a line, at a sufficiently large distance from \(\omega_0\), the intensity distribution corresponds to the statistical theory, and the boundary of this region in general does not depend on \(N\). We note that both effects have been observed: the presence of a statistical wing, and the violation of the linear dependence of the width and shift on \(N\), characteristic of the impact theory, with increasing pressure\(^ {4,30}\).

Unzöld’s arguments are qualitative in character and are based on an extremely simplified scheme. Thus, the shift of the oscillator frequency over the interval \(\rho/v\) (during a collision) is taken to be equal to \(C_n/\rho^n\), and between two collisions—to zero. Thus, instead of a continuous frequency modulation \(x(t)=C_n/[R(t)]^n\), a rectangular one is considered. This circumstance makes Unzöld’s arguments, as was already noted in work\(^ {27}\), insuf-

sufficiently convincing*). Below we shall rely on later published works\(^{28,37,39}\), which make it possible to give a more general and rigorous solution of the question.

Let us return again to oscillator (9). In the most general case, the intensity distribution in the emission line of such an oscillator, \(I(\omega)\,d\omega\), according to (2) and (3), will be determined by the Fourier component:

\[ f_\omega \sim \frac{1}{(2\pi T)^{1/2}} \int_{-T/2}^{T/2} e^{\,i[\eta(t)-(\omega-\omega_c)t]}\,dt . \tag{21} \]

Let us consider (21) separately for large and small values of \(\Delta\omega=\omega-\omega_0\). If \(\Delta\omega\) is large, the integrand in (21) oscillates strongly everywhere except at the points \(t_k\), where

\[ \left(\frac{d\eta}{dt}\right)_{t_k}=\chi(t_k)=\Delta\omega . \]

Recall that \(\chi(t)\) is the shift of the oscillator frequency caused by the perturbation. Therefore the principal contribution to (21) is made by small regions \(\Delta\tau_k\) around these points. Instead of (21) one may write:

\[ f_\omega \sim \frac{1}{(2\pi T)^{1/2}} \sum_k \int_{\Delta\tau_k} e^{\,i[\eta(t)-(\omega-\omega_0)(t-t_k)-(\omega-\omega_0)t_k]}\,dt . \tag{22} \]

Expand \(\eta(t)\) in a series near \(t_k\):

\[ \eta(t)=\eta(t_k)+ \left(\frac{\partial\eta}{\partial t}\right)_{t_k}(t-t_k) +\frac{1}{2}\left(\frac{d^2\eta}{dt^2}\right)_{t_k}(t-t_k)^2+\cdots . \]

\[ \left(\frac{d\eta}{dt}\right)_{t_k}=\Delta\omega , \]

therefore in the exponent in (22) the terms \(\sim(t-t_k)\) cancel, and the series begins with the term

\[ \frac{1}{2}\left(\frac{d^2\eta}{dt^2}\right)_{t_k}(t-t_k)^2 . \]

In the integration, the essential region is \(\Delta\tau_k\), where this term \(\leqslant 1\). (Beyond this, strong oscillations begin.) Hence it is easy to obtain the size of this region:

\[ \Delta\tau_k \approx \sqrt{2}\left(\frac{d^2\eta}{dt^2}\right)_{t_k}^{-1/2} = \sqrt{2}\left(\frac{d\chi}{dt}\right)_{t_k}^{-1/2}. \tag{23} \]

If within this region the next term of the expansion

\[ \frac{1}{6}\left(\frac{d^3\eta}{dt^3}\right)_{t_k}(t-t_k)^3 \ll 1, \]

*) This remark also applies to work \(^{43}\).

for which it is necessary that the relation

\[ \left(\frac{d^3\chi}{dt^3}\right)_{t_k} \left(\frac{d^2\chi}{dt^2}\right)_{t_k}^{-3/2} = \left(\frac{d^2\chi}{dt^2}\right)_{t_k} \cdot \left(\frac{d\chi}{dt}\right)_{t_k}^{-3/2} \ll 1, \tag{24} \]

be satisfied; then the series may be cut off at the term \(\sim (t-t_k)^2\), and in each term of the sum (22) the limits of integration may be extended from \(-\infty\) to \(+\infty\) (outside \(\Delta\tau_k\), because of oscillations, the integration gives zero).

In this case it is easy to obtain\({}^{39}\):

\[ \overline{f_\omega f_\omega^{*}}\,d\omega = \frac{1}{T}\sum_k \left(\frac{d\chi}{dt}\right)_{t_k}^{-1} d\omega . \tag{25} \]

But

\[ \sum_k \left(\frac{d\chi}{dt}\right)^{-1} d\omega \]

is the time during which

\[ \omega_0+\chi(t)-\omega_p \]

is contained in the interval \(\omega,\omega+d\omega\), and (25) gives the statistical distribution. Let us consider relation (24) for small \(N\), when the influence of only a single nearest particle may be taken into account. In this case

Fig. 2.

\[ \chi(t)= \frac{C_n}{\left\{\rho^2+v^2(t-t_0)^2\right\}^{n/2}} . \]

If a small neighborhood around the point \(t_0\) is excluded from consideration, then

\[ \frac{d\chi}{dt}\simeq \frac{C_n v}{\rho^{(n+1)}}, \qquad \frac{d^2\chi}{dt^2}\simeq \frac{C_n v^2}{\rho^{(n+2)}}, \tag{26} \]

and relation (24) takes the form:

\[ \frac{C_n}{v\rho^{(n-1)}}\gg 1. \tag{27} \]

But according to Fig. 2, which gives the graph of \(\chi(t)\), the points \(t_k\) give only those flybys for which

\[ \frac{C_n}{\rho^n}\gg \Delta\omega, \]

or, in other words, we are interested in collisions with

\[ \rho\ll \rho_{\Delta\omega} = \left(\frac{C_n}{\Delta\omega}\right)^{1/n}. \]

Taking this into account, (27) may be rewritten in another form:

\[ \Delta\omega \gg \frac{v^{\,n/(n-1)}}{C_n^{1/(n-1)}} = \Omega . \tag{28} \]

This same result can also be obtained in a somewhat different way[^28]. If in the expansion of $\eta(t)$ one also retains the term $\sim (t-t_k)^3$, then for $I(\omega)\,d\omega$ one can obtain the expression

\[ I(\omega)\,d\omega = \frac{4\pi N C_n^{3/n}}{n(\omega-\omega_0)^{1+3/n}} \left\{ 1-\frac{n}{36}\cdot\left(\frac{2n+1}{2n}\right) \left(\frac{n^2-1}{n^2}\right) \frac{v^2}{(\omega-\omega_0)^{2-2/n}C_n^{2/n}} \right\}, \tag{29} \]

which for $\Delta\omega \gg \Omega$ goes over into the statistical distribution of the intensity in the wing of the line (20). Relation (28) establishes that the region of applicability of the statistical theory is the wing of the line.

Conversely, for the center of the line, where $\Delta\omega \ll \Omega$, the impact approximation is valid. Indeed, in (21) the change of the phase $\eta(t)$ during a collision may be regarded as instantaneous if $1/\Delta\omega$ is much greater than the duration of the collision:

\[ 1/\Delta\omega \gg \rho/v. \]

But, according to impact theory, the principal role in the broadening of the line is played by collisions with
$\rho \leq \rho_0 \simeq \left(\dfrac{C_n}{v}\right)^{\frac{1}{n-1}}$. Substituting $\rho_0$, we obtain
$\Delta\omega \ll \dfrac{v^{n/(n-1)}}{C_n^{1/(n-1)}}$—the relation inverse to (28).

Relations (27) and (28) acquire a simple physical meaning if one approaches them from the standpoint of general spectral regularities under frequency modulation. Let us consider two examples:

1) $x=a\cos\mu t$, $\eta(t)=\dfrac{a}{\mu}\sin\mu t$. The character of the spectrum of the oscillator
$f(t)=e^{i[\omega_0t+\eta(t)]}$ depends essentially on the magnitude of the modulation index $a/\mu$. In the general case the spectrum contains a whole series of components with frequencies that are multiples of $\mu$. Only in the limiting case $a/\mu\gg 1$ does $I(\omega)\,d\omega$ coincide with the statistical distribution, and the width of the spectrum coincides with the swing width $2a$[^29].

2) $x=a_n$ during the interval $t_n<t<t_{n+1}$; $a_n$ and $\tau_n=t_{n+1}-t_n$ are random independent variables. For the segment $n$, $I_n(\omega)\,d\omega$ is a band of width $2/\tau_n$, symmetric with respect to $\omega_0+a_n$. It is easy to see that this segment gives a contribution to the intensity in a small frequency interval around $\omega_0+a_n$, as required by the statistical theory, only if $1/\tau_n\ll a_n$. Here the role of the modulation index is played by $a_n\tau_n$.

Thus, the main role is played not by the “modulation rate” itself (in the first case $\mu$, in the second $1/\tau_n$), but by the ratio of a certain parameter characterizing the depth of modulation to this rate.

In our problem it is natural to take as this parameter the maximum frequency shift \(C_n/\rho^n\) in a collision, and as the modulation velocity \(v/\rho\)—the reciprocal of the collision duration.

In this case (27) is a complete analogue of the relations \(a/\psi \gg 1\) and \(a_n\tau_n \gg 1\), which determine the boundary of applicability of the statistical theory; this analogy was also noted earlier^44. In essence, the same analogy is implicitly taken as the basis of Unsöld’s reasoning^2.

Let us return again to relation (27). According to (5), \(C_n/v\rho^{(n-1)}\), to within a factor of order unity, represents the total phase shift in a collision. Consequently, the principal role in the formation of the statistical wing is played by radiation during strong collisions, i.e. during passages inside the Weisskopf radius \(\rho_0\). This makes it possible to assert that the results obtained above under the assumption of the influence of only the nearest particle are valid as long as the mean distance to the perturbing particle

\[ \bar{R}=\left(\frac{3}{4\pi N}\right)^{1/3} \]

is much greater than \(\rho_0\), or

\[ \rho_0 N^{1/3}\ll 1. \tag{30} \]

Let us now determine under what conditions the greater part of the integral intensity of the line falls in the impact region. Relation (28) makes it possible to answer this question as well. It is easy to see that for this it is necessary that \(\Omega\) considerably exceed the impact width \(\gamma\):

\[ \gamma \simeq 2\pi \rho_0^2 Nv \ll \Omega = \frac{v^{n/(n-1)}}{C_n^{1/(n-1)}} . \]

But, since \(\rho_0 \simeq (C_n/v)^{1/(n-1)}\), the last relation is equivalent to (30).

Thus, at low pressures, as long as (30) is satisfied, impact broadening plays the decisive role—only a relatively negligible part of the total intensity falls in the statistical wing.

Relation (30), to within inessential numerical factors, coincides with the criteria of applicability of the impact theory of S. L. Mandelstam and N. N. Sobolev^40 and Foley^30. The integral character of these criteria is obvious. Indeed, (30) means nothing other than the possibility of neglecting the intensity contained in the statistical wing in comparison with the total intensity of the line. In the spectral approach, i.e. when clarifying the role of the impact and statistical mechanisms of broadening separately for each frequency interval, it is necessary to use the more general relation (28)*).

\[ \text{*) It may happen that the region } \Delta\omega \ll \Omega \text{ is entirely covered by Doppler broadening. In this case, despite the fulfillment of relation (30), only the statistical wing is accessible to observation.} \]

When the pressure is increased, beginning with \(N \simeq 1/\rho_0^3\), inequality (30) is violated. This means that the duration of a collision \(\rho_0/v\) becomes greater than the mean free time \(1/\pi \rho_0^2 Nv\). The assumptions underlying impact theory evidently lose their validity. Whether statistical theory is applicable in this case remains unclear. The answer to this question could be given by condition (24), but now one can no longer use the estimates (26), and the calculation of \(\dfrac{dx}{dt}\) and \(\dfrac{d^2x}{dt^2}\) in the general case of the combined action of many particles is far from trivial. We shall not dwell on this question in greater detail, since at such high densities the principal role is played by the interaction of particles at short distances \((R < \rho_0)\), about which we have only the most tentative ideas.

Let us now turn to Doppler broadening. Usually this effect is treated as purely statistical.^1 The frequency of an oscillator whose velocity component in the direction of the ray of sight is equal to \(v\), in accordance with the Doppler principle, is shifted by the amount \(\omega_0 v/c\). Starting from the Maxwellian velocity distribution

\[ W(v)\,dv=\frac{1}{\sqrt{\pi}} e^{-(v/v_0)^2}\frac{dv}{v_0} \tag{31} \]

and putting \(I(\omega)\,d\omega = W(v)\,dv\), \(\Delta\omega=\omega-\omega_0=\omega_0 v/c\) and \(d\omega=\omega_0/c\,dv\), it is easy to obtain:

\[ I(\omega)\,d\omega= \frac{1}{\sqrt{\pi}} e^{-\left(\frac{\omega-\omega_0}{\Delta\omega_D}\right)^2} \cdot \frac{d\omega}{\Delta\omega_D}, \tag{32} \]

where \(v_0=\sqrt{2\overline{v^2}}\) and \(\Delta\omega_D=\omega_0\dfrac{v_0}{c}\) is the Doppler line width.

In deriving (32) it is assumed that the spectrum of an oscillator moving with radial velocity \(v\) contains only one frequency \(\omega_0(1+v/c)\). This is indeed the case if \(v\) does not change with time. If, however, \(v=v(t)\), then \(I(\omega)\,d\omega\) may differ greatly from (32). In the present case we are dealing with a rectangular frequency modulation (the velocity of the atom is constant between collisions and changes discontinuously at the instant of collision). An analogous example has already been considered above. There the condition for the applicability of the statistical concept was found; in the present case it takes the form \(\dfrac{\omega_0}{c}v_n\tau_n \gg 1\). Substituting for \(v_n\) and \(\tau_n\) the mean values \(\bar v\) and \(\tau_0\) (the mean free time), we obtain

\[ \frac{\omega_0}{c}\bar v \gg 1/\tau_0 \quad \text{or} \quad L \gg \lambda, \tag{33} \]

where \(L=\bar v\tau_0\) is the mean free path, \(\lambda=2\pi c/\omega_0\) is the wavelength of light. In accordance with (33), especially large deviations from (32) should be expected when \(L\gg\lambda\). But in this case \(1/\tau_0\gg\Delta\omega_D\), and collisions begin to play the main role in the broadening of the line. It must be noted that, in general, there are no grounds for separating the effects of interaction and the Doppler effect. Indeed, the loss of coherence in a collision may be caused both by a phase shift and by a change in the velocity of the atom. We shall therefore take both effects into account simultaneously, for which we put

\[ f(t)=e^{i\omega_0 t+i\frac{\omega_0}{c}\int v(t)\,dt+i\int \chi(t)\,dt}. \tag{34} \]

The spectrum of the oscillator (34) is easily obtained in the impact-theory approximation. A collision is accompanied by a change in the velocity of the atom and a phase shift. Between two collisions \(v=\mathrm{const}\) and \(\int \chi(t)\,dt=\mathrm{const}\). It can be shown that in this case \(I(\omega)\,d\omega\) will be determined by the following expression:

\[ I(\omega)\,d\omega= \frac{\gamma}{2\pi^{3/2}v_0} \int_{-\infty}^{\infty} \frac{e^{-v^2/v_0^2}\,dv} {(\omega-\omega_0-\Delta-\omega_0 v/c)^2+(\gamma/2)^2}. \tag{35} \]

A detailed analysis of a distribution of this type may be found in \(^{53,54}\). If the impact width \(\gamma\ll\Delta\omega_D\), then at the center of the line \((\omega-\omega_0-\Delta)\ll\Delta\omega_D\), and (35) coincides with (32). In the wings of the line \((\omega-\omega_0-\Delta)\gg\Delta\omega_D\), the Doppler distribution is replaced by the dispersion wing \(\gamma/2\pi(\omega-\omega_0)^2\). If, on the contrary, \(\gamma\gg\Delta\omega_D\), then (35) passes over into the impact intensity distribution (16). As a rule, the optical cross section \(\pi\rho_0^2\) is greater than the gas-kinetic one; therefore \(\gamma>2/\tau_0\), and the relation \(\Delta\omega_D\gg\gamma\) is equivalent to (33).

For \(L\ll\lambda\) the relation (32) becomes inapplicable, but in this case \(\Delta\omega_D\ll\gamma\), and Doppler broadening in general ceases to play an essential role.*)

We have in mind, however, cases in which Doppler broadening even for \(L\ll\lambda\) is not masked by interaction effects, and the separation of the Doppler effect is fully justified.**)

*) It should be emphasized that (35) was obtained in the impact-theory approximation. If \(\Delta\omega_D\simeq\Omega\) or even greater than \(\Omega\), then a statistical wing directly adjoins the Doppler core of the line.

**) There are a number of indications that, for totally symmetric vibrations of nonpolar molecules (for example, \(\mathrm{C_6H_6}\)), the effective cross section for impact broadening is considerably smaller than the gas-kinetic one \(^{45,56}\). Moreover, the interaction does not affect the shape of the Rayleigh-scattering line in a gas, since the latter is determined by forced, and not natural, vibrations of the oscillator \(^{42,55}\).

Therefore, alongside the general consideration presented above, it is of interest to trace the dependence of the Doppler broadening on pressure separately from the effects of interaction.

Let us again consider oscillator (34), omitting in the exponent the last term, which accounts for the interaction.

For \(L \gg \lambda\), the additional increment of phase during the time between two collisions, due to the Doppler effect, is of order \(\omega_0 \bar v / c\, \tau_0 \gg 1\). In the present case, the Doppler shift of the frequency leads to a loss of coherence in exactly the same way as does the phase shift due to interaction. Therefore the calculation of \(I(\omega)d\omega\) is readily carried out analogously to the way this is done in impact theory\(^1\).

\[ I(\omega)d\omega \sim \sum_n \left| \int_0^{\tau_n} e^{\,i\left[\left(\omega_0-\omega\right)+\frac{\omega_0}{c}v_n\right]t}\,dt \right|^2 d\omega . \tag{36} \]

Averaging (36), after carrying out the integration, over all possible values of \(\tau_n\) and \(v_n\), one easily obtains for \(I(\omega)d\omega\) an expression completely analogous to (35), with the only difference that \(\Delta=0\) and \(\gamma=2/\tau_0\)*).

This result shows that the true intensity distribution differs from (32) by the presence of a dispersion wing \(1/\tau_0(\omega-\omega_0)^2\), with only a small fraction of the total intensity falling in this wing. As the pressure increases, the relative weight of the dispersion wing increases, since \(1/\tau_0\sim N\), but only so long as \(L\) is still greater than \(\lambda\). In the other limiting case, \(L \ll \lambda\), \(\omega_0 \dfrac{\bar v}{c}\tau_0 \ll 1\). Positive and negative values of \(v\) are equally probable; therefore the additional Doppler phase shift, of order \(\omega_0 \dfrac{\bar v}{c}\tau_0\), remains at all times a small quantity, much less than unity. It is obvious that this small additional phase shift cannot lead to a loss of coherence.

The smallness of the phase

\[ \frac{\omega_0}{c}\int v(t)\,dt=\beta(t) \]

makes it possible to decide the question of the form of \(I(\omega)d\omega\) very simply. For \(\beta(t)\ll 1\)

\[ f(t)=\sin\{\omega_0 t+\beta(t)\}\approx \sin\omega_0 t+\beta(t)\cos\omega_0 t \tag{37} \]

and \(I(\omega)d\omega\) splits into two parts: a band, the intensity distribution in which is determined by the form of \(\beta(t)\), and an unbroadened

*) The distribution function in \(\tau\) has the form \(1/\tau_0 e^{-\tau/\tau_0}d\tau\). In averaging over \(v\) we use \(W(v)dv\) from (31).

line (meaning, of course, the absence specifically of Doppler broadening). The intensities contained in the two parts are in the ratio

\[ \overline{\beta^2}:1=\left(\omega_0\,\frac{\bar v}{c}\,\tau_0\right)^2 =\left(2\pi\,\frac{L}{\lambda}\right)^2 . \]

Thus, for \(L \ll \lambda\) the Doppler effect gives rise only to a low-intensity wing with intensity \(\sim 1/N^2\)*). At higher pressures, when the collision duration becomes of the order of \(\tau_0\), it is necessary to take into account the radiation during the collision. In this case the recoil momentum of the photon is distributed between both interacting particles, which somewhat changes the formulation of the problem (as applied to Rayleigh scattering, this question was considered in \(^{42}\)).

IV. ANALYSIS OF THE LAWS OF INTERACTION

1. Time dependence of the perturbation

Above it was assumed that the perturbation changes adiabatically, i.e., so slowly that the collision does not induce transitions between different stationary states of the atom. This assumption is essential from two points of view. First, it made it possible to assume that the perturbation manifests itself only in a change of the oscillator phase, without affecting its amplitude. Second, the absence of transitions within a single level between different \(M_J\)-states makes it possible to consider the broadening of the separate \(\pi\)- and \(\sigma\)-components of the line independently of one another.

The adiabaticity condition is easy to formulate: it is necessary that the collision duration \(\rho/v\) be sufficiently large—so that \(\hbar v/\rho\) be considerably smaller than the energy difference \(\Delta E\) of two stationary states. Since the frequencies of the atomic spectrum are of the order of \(10^{15}\ \mathrm{sec}^{-1}\), this condition is, as a rule, fulfilled**). This is confirmed, in particular, by the fact that the cross section for quenching of resonance fluorescence is much smaller than \(\pi\rho_0^2\). Transitions within a single level require

*) Similar results were obtained by Dicke \(^{41}\), who considered the same question on the basis of the following model: a system of oscillators executes reciprocal-translational motion in a one-dimensional box of length \(L\) with velocities distributed according to (31). In the case of an oscillator whose motion is described by the diffusion equation, Dicke obtained for \(I(\omega)\,d\omega\) a dispersion distribution with width \(4\pi D/\lambda^2\), where \(D\) is the self-diffusion coefficient. Since this result is given without derivation, the reason for the discrepancy remains unclear. We note only that the presence of an unbroadened peak does not depend on the particular form of \(\beta(t)\) and is connected only with the obvious fact of the smallness of the phase

\[ \frac{\omega_0}{c}\int v(t)\,dt \]

for \(L \ll \lambda\).

**) A quantitative treatment of this question is contained in \(^{30}\).

special consideration. Let us first recall the formulation of the problem. In calculating the energy of term splitting, one usually uses a coordinate system with the \(OZ\) axis directed along the perturbing particle. In this case, owing to the axial symmetry of the perturbation (interaction with a spherically symmetric particle is assumed—a charge or an atom in the \({}^{1}S_{0}\) state), the interaction energy does not depend on the coordinates \(x\) and \(y\) of the atomic electron. In the matrix of the coordinate \(z\), only the elements for transitions without change of \(M_J\) are different from zero, and therefore states with different \(M_J\) behave, under the application of perturbation theory, independently of one another. But this coordinate system does not remain fixed in space. During the collision the \(OZ\) axis, following the perturbing particle, turns through an angle of order \(2\pi\). If transitions between different \(M_J\)-states are absent, then the vector of the total angular momentum \(J\) adiabatically follows the \(OZ\) axis and the atom is reoriented in space. If, on the contrary, such transitions are possible, then the orientation of the vector \(J\) in space is preserved.

During the collision the splitting of the level with respect to \(M_J\) is of the order of magnitude \(\hbar C_n/\rho^n\), and the condition of adiabaticity requires that the relation \(C_n/\rho^n \gg v/\rho\) be satisfied, or

\[ \rho \ll \left(\frac{C_n}{v}\right)^{1/(n-1)} \simeq \rho_0 . \tag{38}* \]

Thus, close collisions (flights inside the Weisskopf radius) proceed adiabatically. Therefore the statistical wing of a line produced by radiation during collisions with \(\rho \ll \rho_0\) can be constructed by superposing the individual components with allowance for their relative intensities. It is easy to see that the resulting intensity distribution in this wing will be determined by expression (20), if one puts

\[ C_n^{3/n}=\sum_k I_{0k} C_{nk}^{3/n}\Big/\sum_k I_{0k}, \tag{39} \]

where \(I_{0k}\) are the relative intensities of the components.

In the region of impact broadening, owing to the violation of adiabaticity, the separation of individual components no longer has meaning. In the first approximation one may use a mean value \(\overline{C}_n\) common to the whole line (this is proposed, for example, in \({}^{2}\)). However, a number of difficulties then arise. If all the \(M_J\)-sublevels are shifted

*) This result can also be obtained in a somewhat different way. Rotation of the coordinate system with angular velocity \(\varepsilon\) is equivalent, as is known,\({}^{57}\) to the imposition of an external magnetic field \(H=2mc\varepsilon/e\), in the present case a variable one, since \(\varepsilon\) depends on time. This field contains frequencies of order \(v/\rho\). Obviously, transitions are possible if \(v/\rho \gg C_n/\rho^n\), and impossible if \(v/\rho \ll C_n/\rho^n\), in complete agreement with (38).

...are shifted to one side, then \(\overline{C}_n\) can be determined from the mean shift, or simply set equal to

\[ \overline{C}_n=\sum_k C_{nk} I_{ok}\Big/\sum_k I_{ok}. \]

If, however, the splitting of the level is symmetric, as, for example, in the linear Stark effect, then the mean shift is equal to zero. Does this mean that impact broadening is altogether absent? Obviously not.

In an “adiabatic collision” the atom reorients itself in space through an angle \(2\pi\). The phase shift associated with this rotation, irrespective of the phase shift \(\eta(\rho)\) caused by the change in frequency, leads to a loss of coherence[^2].

Taking (38) into account, one may take as \(\overline{C}_n\) the value \(C_n\) averaged over the absolute value. It must be remembered, however, that such an estimate can give no more than an order of magnitude. It would be more correct to carry out the whole calculation in a coordinate system fixed in space. In this case the interaction energy will depend on all three coordinates of the atomic electron \(x\), \(y\), and \(z\), which makes it necessary to take account of the degeneration with respect to \(M_J\) and to solve the secular equation.

On the basis of a number of results obtained in works \(^{34}\) and \(^{26}\)*), in review \(^{2}\), in complete agreement with what has been set forth above, it is asserted that the differences between exact calculations and the elementary treatment (introduction of the mean value \(\overline{C}_n\)) are small in the case of interactions \(\sim R^{-4}\) and \(R^{-6}\), and reach their greatest value for perturbations \(\sim R^{-2}\) and \(R^{-3}\).

For example, taking account of transitions between different \(M_J\)-states in the case of broadening due to self-pressure \((\sim R^{-3})\) reduces the impact width \(\gamma\) by approximately a factor of 2.

2. Interaction with Charged Particles

A charge \(Q\), situated at a distance \(R\) from the center of the atom, creates at the center of the atom an electric field of strength \(\mathcal{E}=Q/R^2\). It is usually assumed\(^{1,2}\) that over the extent of the atom the field does not change appreciably, retaining the same value as at its center.

The behavior of an atom in a homogeneous electric field has been well studied: the presence of the field leads to a splitting or displacement of the line, for hydrogen proportional to \(\mathcal{E}\) (linear effect

* This apparently very substantial work (judging by the numerous references in \(^{2}\)) unfortunately was not published in the periodical literature available to us.

(Stark), and in the general case \(\sim \mathscr{E}^{2}\) (quadratic Stark effect*).

In the first case the frequency shift \(x \sim R^{-2}\), in the second \(\sim R^{-4}\). Relation (28) of the preceding section makes it possible to elucidate the nature of the broadening produced by ions and electrons. This question is considered in detail in \(^{2}\) as applied to typical conditions existing in stellar atmospheres.

In the case of the linear Stark effect, at a temperature of several thousand degrees (when the concentrations of charged particles are sufficiently large), outside the Doppler width the electrons produce impact broadening, and the ions, often, statistical broadening; moreover, estimates show that under these conditions \(\gamma \ll \Delta\omega_D\), and therefore the influence of the electrons may generally be neglected**).

In the quadratic Stark effect, on the contrary, the main role is played by impact broadening produced by electrons. Interaction with ions somewhat increases the impact width—approximately by 30%, since, according to (17), \(\gamma_i \sim v^{1/3}\). For lines with large values of the interaction constant \(C_4\) (as a rule, for metal lines \(C_4\) is of the order of \(10^{-15}\)–\(10^{-12}\ \mathrm{cm}^4/\mathrm{sec}\)) the appearance of a statistical wing produced by ions is possible. For example, for the broadening of the line Mg \(5528\ \text{\AA}\) \((3^{1}P_1 - 4^{1}D_2)\) by ions \(\mathrm{H}^{+}\), \(C_4 = 5\cdot 10^{-13}\ \mathrm{cm}^4/\mathrm{sec}\) at \(T = 5000^\circ\mathrm{K}\), \(v = 10^{6}\ \mathrm{cm}/\mathrm{sec}\), and \(\Omega = v_i^{4/3}/C_4^{1/3} \simeq 10^{12}\ 1/\mathrm{sec}\).

The Doppler width of the line at this temperature is \(\Delta\omega_D \simeq 2\cdot 10^{10}\).

Since the energy of splitting of a term in the quadratic Stark effect does not depend on the sign of \(M_J\), all levels with both positive and negative \(M_J\) are shifted in one direction.

As a consequence, the statistical wing is situated on one side of the line center, namely the short-wavelength side (if \(C_4 > 0\)) or the long-wavelength side (if \(C_4 < 0\)).

We shall not dwell on consideration of various concrete cases of broadening (see on this point \(^{2}\)), but shall turn to the considerably more interesting question of the extent to which the assumption of field homogeneity corresponds to reality.

The interaction energy of an atom with a point charge \(Q\), located at a distance \(R\) from the center of the atom in the direction \(OZ\), can be represented in the form of an expansion in powers of \(r\) (the coordinate of the atomic electron):

\[ u = u' + u'' + \ldots = -eQ\{z/R^2 + (r^2 - 3z^2)/2R^3 + \ldots\}. \tag{40} \]

* Similar to hydrogen, in sufficiently strong fields other atoms located in hydrogen-like (as a rule, strongly excited) states also behave in this way.

** Analogous results were obtained by Spitzer in papers \(^{32, 33, 34}\), specially devoted to the broadening of hydrogen lines.

Here the first term is the dipole interaction, the second the quadrupole interaction, etc. Assuming the field to be homogeneous, we took into account only the first term of this expansion, which is most significant at large \(R\). Let us determine what allowance for the subsequent terms can give for the hydrogen atom. A hydrogen atom in a state with principal quantum number \(n\) has linear dimensions \(\sim a_0 n^2\), where \(a_0=\hbar^2/me^2\) is the atomic unit of length. Consequently, the first term in expression (40) is of order \(eQ a_0 n^2/R^2\), and the second is \(eQ a_0^2 n^4/R^3\). All terms in (40) connected with the inhomogeneity of the field decrease, as \(R\) increases, faster than \(1/R^2\); therefore the inhomogeneity of the field has an effect only when the particles approach closely, i.e. in the wings of the line. The influence of the additional terms begins to show itself when the second term becomes equal in order of magnitude to the first, i.e. for \(R\lesssim a_0 n^2\), or
\[ \omega-\omega_0 \gtrsim eQ/a_0\hbar n^2 = 4.1\cdot 10^{14}/n^2 . \]
This estimate shows that neglect of the field inhomogeneity for the first members of a series (small \(n\)) does not lead to large errors and is therefore quite permissible (this question is discussed in more detail in \(^{34}\)).

For non-hydrogen-like levels the situation is considerably more complicated. In this case the correction of the first approximation to the level energy \(u'_{nn}\sim 1/R^2\) is equal to zero. The second-order correction due to \(u'\), corresponding to the quadratic Stark effect in a homogeneous field, \((\Delta E')\), is proportional to \(e^2Q^2/R^4\), whereas the quadrupole splitting of the level (the second term in (40)) is \(\Delta E''\sim eQ/R^3\). Consequently, this quadrupole interaction may play the principal role in line broadening.

According to the calculations given in work \(^{46}\), the quadrupole splitting of a term for an atom with one valence electron is determined by the expression
\[ \Delta E''_{njm}=-\frac{eQ}{R^3}\rho_{nl}\frac{j(j+1)-3m^2}{4j(j+1)}, \tag{41} \]
where \(j\) is the total angular momentum, \(l\) the orbital angular momentum, and \(m\) the projection of \(j\) on the \(OZ\) axis. \(\rho_{nl}=\int R_{nl}^{2}(r)r^4\,dr\) is the mean value of \(r^2\) in the \((n,l)\)-state. As is seen from (41), the character of the splitting depends essentially on \(j\). In the general case, with the exception of \(j=3/2\), the splitting of the term is asymmetric. The term \(j=3/2\) is split symmetrically:
\[ \Delta E''_{nj=3/2,\;m=\pm 3/2}=\frac{eQ}{5R^3}\rho_{nl} \quad\text{and}\quad \Delta E''_{nj=3/2,\;m=\pm 1/2}= \]
\[ =-\frac{eQ}{5R^3}\rho_{nl}. \]
For the term \(j=1/2\) the quadrupole effect is altogether absent: \(\Delta E''=0\).

According to (18), for an interaction \(\sim 1/R^3\) the line width \(\gamma=2\pi^2 C_3 N\) does not depend on \(v\); therefore ions and electrons play the same role in the impact broadening of the line. In exactly the same way, the general expression for the line shift \(\Delta=N v \sigma_1\) does not depend on \(v\), since in this case \(\sigma_1\sim 1/v\) (this is readily obtained from (14), using (5)).

Thus, the magnitude of the shift is the same for ions and electrons, while the direction is different, since \(\Delta E''_{n/m}\) depends on the sign of \(Q\). Therefore, for equal densities of ions and electrons the total line shift is absent, despite the asymmetry of the splitting*).

It is obvious that the role of quadrupole splitting is especially great when the quadratic Stark effect is small. For example, for the resonance line Ca \(\lambda=4227\ \text{\AA}\) \((4s^2\,{}^1S_0-4s4p\,{}^1P_1)\) the constant of the quadratic Stark effect is \(C_4=-0.78\cdot 10^{-15}\ \text{cm}^4/\text{sec}\), and at \(T\approx 5000^\circ\mathrm{K}\)

\[ \gamma_4=\gamma_{4\mathrm{el}}+\gamma_{4\mathrm{ion}} =11.4\,C_4^{2/3}N\left(v_{\mathrm{el}}^{1/3}+v_{\mathrm{ion}}^{1/3}\right) =4.4\cdot 10^{-7}N, \]

where \(N=N_{\mathrm{el}}=N_{\mathrm{ion}}\).

As was shown in paper \(^{46}\), on the basis of the general expression for quadrupole splitting, in the case of several valence electrons, for the level \(4s^2\,{}^1S_0\), \(\Delta E''=0\), while for the level \(4s4p\,{}^1P_1\)

\[ \Delta E''_{j=1,\ m=\pm 1}=\frac{69}{5}\,eQ\,a_0^2/R^3 \quad\text{and}\quad \Delta E''_{j=1,\ m=0}=-\frac{138}{5}\,eQ\,a_0^2/R^3 . \]

Hence one can readily obtain \(C_3\approx 10^{-7}\ \text{cm}^3/\text{sec}\), \(\gamma_{3\mathrm{el}}=\gamma_{3\mathrm{ion}}=2\pi^2 C_3N=2\cdot 10^{-6}N\), and the total width \(\gamma=4\cdot 10^{-6}N\). Thus, in the present case the quadrupole Stark effect in an inhomogeneous field plays the decisive role.

Fig. 3.

In the example considered, \(\gamma_3\gg\gamma_4\), and \(\gamma_4\) may be neglected. In the case \(\gamma_3\approx\gamma_4\), the two effects cannot simply be added, since the shifts of the levels may have different signs.

Let us examine this in more detail using the example of the broadening of the resonance doublet Na \(\lambda=5890\text{--}96\ \text{\AA}\) \((3s^2S_{1/2}-3p^2P_{3/2,\ 1/2})\).

For the line \(\lambda\,5896\ (S_{1/2},\,P_{1/2})\) the quadrupole effect is absent, \(C_4=3\cdot 10^{-15}\ \text{cm}^4/\text{sec}^{58}\), in accordance with which \(\gamma_4=1.1\cdot 10^{-6}N\), and the line shift is \(\Delta_4=-0.95\cdot 10^{-6}N\) (again at \(T=5000^\circ\mathrm{K}\)). The splitting of the terms \(P_{3/2}\) and \(S_{1/2}\) in a uniform electric field is shown in Fig. 3; the possible transitions are also shown there: \(1/2\to 1/2\), \(-1/2\to -1/2\) (\(\pi\)-components), \(1/2\to -1/2\), \(-1/2\to 1/2\) (\(\sigma\)-components), and \(3/2\to 1/2\), \(-3/2\to -1/2\) (\(\sigma'\)-components).

*) In \(^{47}\) an attempt was made to explain the shift of Na lines in an arc by the influence of field inhomogeneity. As is clear from the above, it is precisely for the line shift that the field inhomogeneity is immaterial.

The splitting constants according to \({}^{58}\) are equal to

\[ C_4(\pi)=+4.3\cdot 10^{-15}\ \mathrm{cm^4/sec},\qquad C_4(\sigma)=-4.3\cdot 10^{-15}\ \mathrm{cm^4/sec} \]

and

\[ C_4(\sigma')=-1.56\cdot 10^{-15}\ \mathrm{cm^4/sec}. \]

The quadrupole splitting is shown in Fig. 4 (the perturbing particle is an electron; for an ion the levels \(+\frac12\) and \(+\frac32\) interchange places). For the term \(3p^2P_{3/2,\,1/2}\), \(\rho_{n1}=41a_0^2{}^{46}\), and from (41) it is easy to obtain the frequency shift of the \(\pi\)-, \(\sigma\)- and \(\sigma'\)-components:

\[ \chi(\pi)=\pm 5\cdot 10^{-8}/R^3,\qquad \chi(\sigma)=\pm 5\cdot 10^{-8}/R^3 \]

and

\[ \chi(\sigma')=\mp 5\cdot 10^{-8}/R^3— \]

the upper sign corresponds to electrons, the lower to ions.

If only the homogeneous field is taken into account, it is easy to obtain*):

\[ \gamma_4=1.15\cdot 10^{-6}N. \]

If, conversely, only the quadrupole splitting is taken into account, then

\[ \gamma_3=2\cdot 10^{-6}N. \]

With the simultaneous allowance for interactions of both types,

\[ \chi=\pm 5\cdot 10^{-8}\cdot R^{-3}-3.4\cdot 10^{-15}\cdot R^{-4}. \tag{42} \]

Fig. 4.

Let us return to the general expression for the line width \(\gamma=2N\overline{v}\sigma_2\), where, in calculating \(\sigma_r\), in (13) we substitute, in accordance with (5) and (6),

\[ \eta(\rho)=2\cdot\frac{\overline{C_3}}{v\rho^2}+\frac{\pi}{2}\cdot\frac{\overline{C_4}}{v\rho^3}. \tag{43} \]

In a collision with an electron the first and second terms in (43), in accordance with (42), have different signs, which leads to mutual compensation and to some decrease of \(\eta(\rho)\). The function \(\eta(\rho)\) for this case is given in Fig. 5. In Section I it was shown that, for an interaction \(\sim R^{-3}\) and \(\sim R^{-4}\), \(\sigma_r=\pi\rho_0^2\), where \(\rho_0\) is determined by the condition \(\eta(\rho_0)=0.64\). In an analogous way one can estimate \(\sigma_r\) in the present case as well. In Fig. 5 the decrease of \(\rho_0\) due to the superposition of perturbations is clearly seen (\(\rho_0=0.38\cdot 10^{-7}\ \mathrm{cm}\) instead of \(\rho_0=0.60\cdot 10^{-7}\ \mathrm{cm}\) for a quadrupole Stark effect alone).

*) Taking into account the remarks concerning the choice of \(\overline{C_n}\) made in the preceding paragraph, we put \(\overline{C_4}=-3.4\cdot 10^{-15}\ \mathrm{cm^4/sec}\) and \(C_3=5\cdot 10^{-8}\ \mathrm{cm^3/sec}\). With this choice, the constant \(\gamma_3\) may turn out to be overestimated by approximately a factor of \(1.5\)–\(2\).

Upon collision with an \(H^+\) ion, conversely, the superposition of perturbations causes an increase in \(\eta(\rho)\) and, consequently, in \(\rho_0\). Let us give the final results of the calculations:

\[ \gamma_{\mathrm{el}} = 0.4 \cdot 10^{-6} N,\qquad \gamma_{\mathrm{ion}} = 1 \cdot 10^{-6} N,\qquad \gamma = \gamma_{\mathrm{el}} + \gamma_{\mathrm{ion}} = 1.4 \cdot 10^{-6} N. \]

If, however, one simply adds \(\gamma_3\) and \(\gamma_4\), one obtains \(3.15 \cdot 10^{-6}N\). Thus, in the general case summation of the widths \(\gamma_3\) and \(\gamma_4\) is inadmissible*).

Fig. 5.

If both effects have the same order of magnitude, then the estimate of \(\rho_0\) must be carried out on the basis of expression (43).

Allowance for the quadrupole effect may also prove substantial in estimating the broadening of ion lines by neutral hydrogen. Upon collision, the ion induces in the hydrogen atom, which is in the normal \(1s\) state, a dipole moment\({}^{22}\)

\[ \mathbf{d} = \frac{9}{2} a_0^3 Q\,\mathbf{R}/R^3. \]

This dipole moment, in turn, creates a field with potential \(\varphi = (\mathbf{d}\mathbf{R})/R^3\) and strength \(\sim 1/R^5\).

*) The influence of field inhomogeneity on the broadening of the resonance doublet by electrons and ions was considered in \([46]\). In that work the line width was assumed equal to \(\gamma_3+\gamma_4\), as a result of which the value of \(\gamma\) for the component \((S_{1/2} - P_{3/2})\) proved to be overestimated by approximately a factor of 3.

The quadratic Stark effect in such a field is \(\sim 1/R^{10}\) and does not play an essential role. The quadrupole splitting, however, is \(\sim 1/R^6\) and must be taken into account.

The expression for the splitting energy is easily obtained by substituting in (41) \(3d_z/R^4 = 27a_0^3Q/2R^6\) instead of \(Q/R^3\). The estimate carried out in [46] shows that the interaction constants \(C_6\) are, in order of magnitude, equal to \(10^{-32}\div 10^{-31}\ \dfrac{\mathrm{cm}^6}{\mathrm{sec}}\).

For example [46], for the line \(\lambda 3933\,\text{\AA}\ \mathrm{Ca}^+\), \(C_6=6\cdot 10^{-32}\ \dfrac{\mathrm{cm}^6}{\mathrm{sec}}\). Under the conditions of the solar atmosphere (hydrogen atom density \(N_0=10^{17}\)) this gives, for the line widths, \(\gamma \simeq 10^9\), and shifts \(\Delta \simeq 3.2\cdot 10^8\), i.e., quantities of approximately the same order as the relativistic ones (the red gravitational shift).

3. Van der Waals Interaction of Neutral Atoms

Figure 6 shows the typical form of the Franck–Condon potential curves representing the initial and final terms of the emitting atom as functions of the distance \(R\) to the perturbing particle. At present we have at our disposal neither theory nor experimental methods that would make it possible to determine the exact course of these curves. London’s dispersion formula describes with sufficient accuracy only the interaction of atoms in normal states at large \(R\). For excited states one must be limited to approximate estimates. In a number of cases even the qualitative course of the curve remains unclear. Thus, for strongly excited states \(V(R)\) may have no minimum. The expansion of \(V(r)\) in powers of \(1/R\) begins with a term \(\sim 1/R^6\). Therefore it is usually assumed that

\[ \Delta\omega=C_6/R^6, \]

discarding all subsequent terms of the expansion. (In Fig. 6 this corresponds to the dashed continuation of the curves.) An approximate estimate\(^4\) gives for the interaction constant \(C_6\) an order of magnitude \(10^{-30}\div 10^{-32}\ \dfrac{\mathrm{cm}^6}{\mathrm{sec}}\). It is obvious that this approximation is valid only in the case when the principal role is played by the interaction

at large distances. Numerous experiments show that impact broadening of the line plays the principal role up to pressures of several tens of atmospheres.^4 This is in agreement with the criterion, obtained in Section III, for the applicability of the impact theory (30). Indeed, for \(C_6 \simeq 10^{-31}\ \mathrm{cm}^6/\mathrm{sec}\) and \(v \simeq 5\cdot 10^4\ \mathrm{cm}/\mathrm{sec}\),

\[ \rho_0 \simeq \left(\frac{C_6}{v}\right)^{\frac{1}{n-1}} \simeq 10^{-7}\ \mathrm{cm}. \]

The mean distance

\[ \overline{R}=\left(\frac{3}{4\pi N}\right)^{1/3} \]

becomes of the order of \(\rho_0\) at \(N \simeq 10^{21}\), which corresponds to pressures of the order of 40–50 atmospheres. According to (18), for \(n=6\) the ratio of the line width to the shift is \(\gamma/\Delta=2.8\), and Fig. 6 shows that \(C_6<0\) and the line is shifted toward lower frequencies (red shift). Fulfillment of this relation may serve as a good test of the \(1/R^6\) law and of formula (43). The experimental data collected in Refs. 4 and 30, and also 18, show that, as a rule, it is precisely the red shift that is observed). The ratio \(\gamma/\Delta\) in a number of cases is close to 2.8. Moreover, in the broadening of the \(D\)-lines by argon, Minkowski observed^4 a red wing with an intensity decrease according to the law \(I(\omega)\sim(\omega-\omega_0)^{-3/2}\), which also agrees with (20) and (43)*).

However, not infrequently \(\gamma/\Delta\) differs noticeably from 2.8 (by 20–30, and sometimes by 100%) in either direction. In a number of cases (though comparatively rare ones) a blue shift was observed instead of a red one.^4 All this indicates the necessity of refining the law of interaction, especially at high densities. Indeed, at pressures of several tens of atmospheres the mean distances between atoms \(\overline{R}\) and \(R_0\) (see Fig. 6) have the same order of magnitude. Therefore one may expect that the inner parts of the curves \(V(R)\), where (43) is inapplicable, begin to play an essential role.

Recently, papers devoted to this question have begun to appear. Thus, in Ref. 54 it was shown that one can obtain substantially better agreement with experiment^49,50,52 if the statistical theory is constructed on the basis of an interaction

\[ A/R^6 + B/R^{12}. \]

Such a refinement has no theoretical value, since the constants \(A\) and \(B\) are chosen purely empirically; it merely emphasizes the inadequacy of (43). Considerably more interesting in this

*) In order of magnitude this shift is close to the relativistic (gravitational) shift and must be taken into account in measuring the latter.

**) According to (28), in the present case the statistical wing begins to appear at \(\omega-\omega_0 \gg v^{6/5}/C_6^{1/5}\). For \(T=300^\circ\mathrm{K}\), \(v\simeq 5\cdot 10^4\) and \(v^{6/5}/C_6^{1/5}\simeq 5\cdot 10^{11}\ \mathrm{sec}^{-1}\).

For large \(T\), for example in stellar atmospheres, the whole line falls in the region of impact broadening.^2

—in the sense of the question of the different broadening and shift of individual components of multiplets. This difference was found in a series of experimental works \(^{15-17,19,51}\).

For example, for the resonance doublet of Rb under the action of A, He, and Ne, the collisional broadening of the component \((S_{1/2}-P_{3/2})\) is greater than that of the component \((S_{1/2}-P_{1/2})\). The reason for this difference was clarified in work \(^{20}\). In calculating the forces of attraction or, in other words, in calculating the curves \(V(R)\) in the region \(R>R_0\), the presence of a quadrupole moment in the \(P_{3/2}\)-state was taken into account. This gave a difference in the widths of the doublet components by a factor of 1.12, which agrees with experiment.

4. Broadening in a homogeneous gas.

(Self-pressure.)

It has been noted repeatedly \(^{1,8}\) that, with an increase in the density of a homogeneous gas, the lines broaden considerably more strongly than when a foreign gas is added. This fact was at first interpreted in favor of the existence of a special broadening, specific to identical atoms, caused by dipole interaction. Such broadening received the name of the coupling width. According to Holtsmark, the coupling width of the emission line of a system of \(N\) identical coupled oscillators (such a system in the general case has \(N\) frequencies) is determined by the mean square deviation of the frequency of the system from the frequency of the unperturbed oscillator, averaged over all possible positions of the oscillators relative to one another. This point of view was subjected to detailed criticism by Weisskopf \(^{1}\) and by Vlasov and Fursov \(^{8}\). They showed that in the given case as well (at not very high pressures) the main role is played by the interaction of two atoms at close approach, i.e., by the collisional mechanism of broadening.

On the basis of the general considerations on phase shift developed by him, Weisskopf gives an estimate of the effective cross section and of the line width. This gives the following values of the interaction constant and of the line width \(^{1}\):

\[ C_3=\frac{e^2}{4m\omega_0}\,f \quad \text{and} \quad \gamma_3=\frac{\pi^2}{2}\,\frac{e^2}{m\omega_0}\,f\cdot N, \tag{44} \]

where \(e\) is the charge of the electron, \(m\) its mass, and \(f\) the oscillator strength of the corresponding line. Vlasov and Fursov interpret the broadening mechanism somewhat differently. In a collision, a resonant exchange of an excitation quantum between the interacting atoms is possible, or, in the language of classical representations, a transfer of energy from one oscillator to another. In this process the coherence of the radiation is disturbed, since the phases of the oscillations of the colliding—

interacting oscillators are different (if only as a consequence of the Doppler shift of the frequency, which is different for the two atoms*). The magnitude of the transferred energy \(\varepsilon\) depends on \(\rho\). Equating \(\varepsilon(\rho)\) to the vibrational energy of the oscillator, one can determine the effective collision radius \(\rho_0\). (Thus, a collision is taken to mean a passage in which all the vibrational energy is transferred from one oscillator to the other.) The calculations carried out in \({}^{8}\) give

\[ \rho_0=\sqrt{\frac{e^2}{m\omega_0}\cdot f\frac{1}{v}} \quad\text{and}\quad \gamma'=2Nv\pi\rho_0^2 =2\pi\cdot\frac{e^2}{m\omega_0}\,f\cdot N. \tag{45} \]

For passages outside the effective radius \((\rho>\rho_0)\), only part of the vibrational energy is transferred. This additional loss of energy, equal during the time \(dt\) to

\[ dE=2\pi Nv\,dt\int_{\rho_0}^{\infty}\rho\varepsilon(\rho)\,d\rho, \]

leads to an additional damping of the vibrations of the atomic oscillator and, consequently, to an additional broadening of the line. This broadening, according to \({}^{8}\), is determined by the expression

\[ \gamma''=\frac{2\pi}{3}\cdot\frac{e^2}{m\omega_0}\,fN, \]

which gives the total width

\[ \gamma=\gamma'+\gamma''=\frac{8\pi}{3}\,\frac{e^2}{m\omega_0}\cdot f\cdot N, \tag{46} \]

differing from (44) only by the factor \(16/3\pi\).

Let us note that only as a result of the incoherence of the vibrations of the interacting oscillators can this loss of energy be interpreted as damping analogous to that caused by the transfer of energy to external degrees of freedom or by radiation in another frequency interval. Thus, in the present case as well, the decisive factor is the disturbance of the phase during a collision.

The difference from the general impact theory developed in Section I consists only in a somewhat different approach to the determination of the effective collision cross section, more appropriate to the present concrete case. More distant passages are also taken into account somewhat differently**. Since the difference between (46) and (44) is very small, in concrete calculations one may use either method.

* Let us recall that this statement is valid for \(L\gg\lambda\), i.e. at sufficiently low pressures (see above). At higher pressures the question requires special consideration.

** From this point of view there are no grounds for opposing this method to Weisskopf’s method, as is done in \({}^{8}\) (this circumstance was already noted in \({}^{9}\)).

Subsequently, on the basis of the method of Vlasov and Fursov, in \(^{9}\) a calculation was made of the widths of the components of multiplets (the collision of two atoms in states with total angular momenta \(J\) and \(J'\) was considered). The calculation gave the same broadening of all members of the multiplets, which is in obvious contradiction with experiment \(^{14, 15, 24, 25, 59}\).

The reason for this discrepancy, as was pointed out in \(^{30}\), where an analogous problem was considered, lies in an incorrect averaging of the interaction.

In \(^{30}\) the Weisskopf method was taken as the basis of the calculation, and the line width, in accordance with (16), was taken to be equal to \(\gamma_3 = 2\pi^2 \overline{C}_3 N\). The mean value of the interaction constant \(\overline{C}_3\) was determined from the mean value of the absolute magnitudes of the splitting energies (see § 1 of this section). In contrast to \(^{9}\), the calculation gave different broadening of the components of multiplets. In particular, for the components of the resonance doublets of the alkali metals the ratio of the widths \(\gamma(S_{1/2} - P_{3/2})/\gamma(S_{1/2} - P_{1/2})\) is equal to \(f\sqrt{2} \simeq \sqrt{2}\). Experiment gives, for Na—1.55, K—1.00, Rb—1.75, and Cs—1.72 \(^{14, 15, 24, 29}\).

As can be seen, the scatter of the experimental data is rather considerable, which makes comparison with theory difficult. Moreover, an indication has recently appeared that this ratio may depend substantially on the experimental conditions. Thus, the ratio of the widths of the components of the Na doublet measured in \(^{59}\) turned out to be 1.14 at pressure \(p = 0.013\) mm Hg and \(T = 300^\circ\mathrm{C}\), and 1.45 at \(p = 0.503\) mm Hg and \(T = 415^\circ\mathrm{C}\).

Somewhat special in the present case is the question of the boundary between the impact and statistical regions of broadening. The point is that the statistical theory in the case \(n = 3\), just like the impact theory, gives in the wing of the line an intensity distribution \(\sim(\omega-\omega_0)^{-2}\).

Nevertheless, the difference between the impact and statistical theories is preserved in the present case as well, since for \(\Delta\omega \ll \Omega\) and \(\Delta\omega \gg \Omega\) the interaction constants are determined differently (see § 1 of the present section). In particular, one may expect different regularities in the broadening of the components of multiplets. For example, calculations carried out on the basis of the statistical theory gave the following expression for the statistical wing \(^{10}\) (only the results of the calculation are given below):

\[ I(\omega)\,d\omega = 2\pi\mu\,\frac{e^2}{M\omega_0}\, f\cdot N \cdot \frac{d\omega}{(\omega-\omega_0)^2}. \]

For the transitions \(({}^{1}S_0 - {}^{1,3}P_1)\), \(({}^{2}S'_{1/2} - {}^{2}P_{1/2})\), and \(({}^{2}S_{1/2} - {}^{2}P_{3/2})\), \(\mu = 2/9,\ 1/3\), and \(0.256\), respectively.

For resonance doublets of the alkali metals \(C_3\) has the order of magnitude \(e^2 f/4m\omega_0 \simeq 10^{-8}\ \mathrm{cm}/\mathrm{sec}\), therefore at \(T = 300^\circ\mathrm{K}\), \(\Omega \simeq 10^{11}\ \mathrm{sec}^{-1}\). Since the Doppler width at this

temperature \(\Delta\omega_D \approx 10^{10}\ 1/\mathrm{sec}\), it may turn out that the experimental data correspond precisely to the intermediate region \(\Delta\omega \sim \Omega\), where neither the impact nor the statistical theory is applicable. This circumstance must be taken into account in interpreting experimental material.

V. QUANTUM-MECHANICAL JUSTIFICATION OF THE THEORY

Let us first recall, in general outline, the course of Weisskopf’s reasoning\(^{1,3}\), which clarified under what assumptions the distribution of intensity in an atomic emission line is determined by relations (2) and (3).

When an atom moves in the field of action of perturbing particles, its electronic terms are functions of the coordinate of the nucleus \(\mathbf R\): \(V_n = V_n(\mathbf R)\). The total energy of the atom \(E\) is equal to the sum of \(V_n(\mathbf R)\) and the kinetic energy of the translational motion of the atom \(\varepsilon\), while the eigenfunction is the product \(\varphi_n(q)\psi_n(\mathbf R,E)\), where \(\varphi_n(q)\) is the eigenfunction of the electron in state \(n\), \(q\) is the coordinate of the electron, and \(\psi_n(\mathbf R,E)\) is the function of the translational motion.

The intensity of radiation in the transition \(n \to n'\) and with simultaneous change of \(\varepsilon\) to \(\varepsilon'\) is proportional to

\[ \left|\iint q\varphi_{n'}^{*}(q)\varphi_n(q)\psi_{n'}^{*}(\mathbf R,E')\psi_n(\mathbf R,E)\,dq\,d\mathbf R\right|^2 . \tag{47} \]

In this transition, according to the law of conservation of energy, a quantum is emitted

\[ \hbar\omega = V_n - V_{n'} + \varepsilon - \varepsilon' = \hbar\omega_p + \varepsilon - \varepsilon' . \tag{48} \]

Suppose that the matrix element \(\int q\varphi_{n'}^{*}(q)\varphi_n(q)\,dq\) is equal to the same value \(A_{nn'}\) as in the absence of perturbation (this corresponds to the assumption of constant amplitude). Then for \(I(\omega)\) we obtain

\[ I(\omega)\sim \left|A_{nn'}\int \psi_{n'}^{*}(\mathbf R,E')\psi_n(\mathbf R,E)\,d\mathbf R\right|^2 . \tag{49} \]

We shall now assume that the translational motion of the atom is quasi-classical, and for simplicity we restrict ourselves to the one-dimensional case. In this approximation

\[ \psi_n(x,E)=\frac{c}{\sqrt p}e^{i/\hbar\int p\,dx}, \]

where the momentum \(p\) is related to the energy \(E\) by the relation

\[ p=\sqrt{2m\varepsilon}=\sqrt{2m(E-V_n)}. \]

Further, \((p-p')v=\varepsilon-\varepsilon'\). Taking (48) into account, one may also write \(v(p-p')=\hbar\omega-\hbar\omega_p\). Bearing in mind that \(dx=v\,dt\), we obtain instead of (49)

\[ I(\omega)\sim \left|A_{nn'}\int e^{i\omega t-i\int\omega_p dt}\,dt\right|^2 . \tag{50} \]

Expression (50) is completely equivalent to relations (2) and (3) for a classical oscillator with constant amplitude and a variable instantaneous frequency \(\omega_p(t)\). The calculation carried out makes clear the mechanism of broadening of spectral lines. To elec-

...the continuous spectrum of the translational motion of the atom adjoins the electronic term on both sides. If the perturbation is absent, then the eigenfunctions of the translational motion are orthogonal and the optical transition is not accompanied by a change in \(\varepsilon\). If, however, the atom is subjected to a perturbation, then this orthogonality is violated and transitions with a change in \(\varepsilon\) become possible. The transfer of part of the electron excitation energy into the kinetic energy of the nuclei also causes a broadening of the lines.

In a series of papers\(^{35-38}\) Jablonski constructs a theory of the broadening of spectral lines on a somewhat different basis.

According to Jablonski, all the gas contained in the given volume forms a single system, similar to a gigantic molecule, with stationary states of nuclear motion. The broadening of the line, just as in Weisskopf’s treatment, is explained by the fact that in an optical transition both the electronic state and the state of translational motion of the atoms change simultaneously. It is obvious that the formulation of the problem in Weisskopf and in Jablonski is almost the same. In the second case only the eigenfunctions of the motion of the atoms are defined somewhat differently. Weisskopf carries out the further calculation of (50) in the impact-theory approximation. Jablonski, however, starting from an integral analogous to (49), obtains the statistical distribution (20)\(*\). On the basis of this divergence in the results, Jablonski comes to the conclusion that the method of Fourier analysis is erroneous in principle and that from (50) it is impossible to obtain the correct expression for \(I(\omega)\,d\omega\). It is easy to show that there are no grounds for such an assertion, since the noted difference in the results is connected not with the formulation of the problem but with the method of calculation. Indeed, expression (50) contains, as two limiting cases, both the statistical distribution (20) and the impact distribution (see Sec. III). Moreover, in the quasiclassical approximation Jablonski’s initial expression for \(I(\omega)\,d\omega\) is completely equivalent to expression (50) (this was shown in \(^{30}\)). To this it should be added that the assumptions underlying Jablonski’s calculations\(^{37}\) practically coincide with the criterion of applicability of the statistical theory.

Weisskopf’s results can also be obtained by a completely different route, taking into account the influence of the surrounding particles by introducing into the Hamiltonian function for the atom the perturbation \(V(t)\). If this perturbation changes sufficiently slowly (adiabatically), then both Dirac’s theory of radiation\(^{30}\) and the correspondence principle\(^{48}\) give for \(I(\omega)\,d\omega\) an expression analogous to (50).

It is easy to see that this route gives nothing new in comparison with \(^{3}\), since here too the principal point is the assumption of the quasiclassical character of the relative motion of the colliding—

\(*\) To within unessential additive terms.

ing particles. Indeed, the transition from a perturbation depending on the coordinates of the nuclei to a perturbation depending on time is possible only in the quasiclassical approximation, when the coordinates of the nuclei may be regarded as prescribed functions of time, and not as dynamical variables.

Summarizing all that has been said above, one may assert that in the quasiclassical approximation the situation is relatively simple—the calculation of \(I(\omega)\,d\omega\) in one way or another reduces to the Fourier analysis of the oscillations of the equivalent oscillator. (The same remark also applies to the works \(^{32-44}\).)

The condition of quasiclassicality is easy to formulate \(^{22}\).

Motion in a centrally symmetric field\(^*\) is quasiclassical for large values of the angular momentum \(\hbar\sqrt{l(l+1)}\), namely, if the condition \(l \gg 1\) is satisfied. Since in this case
\[ \hbar\sqrt{l(l+1)} \approx \hbar l \simeq \mu v \rho, \]
where
\[ \mu=\frac{m_1m_2}{m_1+m_2} \]
is the reduced mass and \(v\) is the relative velocity of the colliding particles, this condition is equivalent to the requirement that the de Broglie wavelength
\[ \lambda=\frac{\hbar}{\mu v} \]
be small in comparison with \(\rho\). If the perturbation is created by molecules or atoms, then this condition is always satisfied (a special consideration may be needed only for the lightest atoms—H and He—at low temperatures). For example, for \(\mu \approx 10^{-22}\,g\) and \(v \approx 5\cdot10^{4}\,\mathrm{cm/sec}\), \(\hbar/\mu v \approx 2\cdot10^{-10}\,\mathrm{cm}\), whereas \(\rho_0\) has the order of magnitude \(10^{-8}\div10^{-7}\,\mathrm{cm}\).

The situation is considerably more complicated if the perturbation is produced by electrons, for which even at high temperatures the quasiclassical approximation, as a rule, turns out to be inapplicable. For example, at \(T \approx 5000^\circ\mathrm{K}\),
\[ \frac{\hbar}{\mu v}\approx 3\cdot10^{-8}\,\mathrm{cm}. \]
Here, however, help is provided by the circumstance that electrons always produce purely impact broadening (see § 2 of Section IV).

In the approximation of impact theory, the calculation of the integral entering into (49) is considerably simplified. In integrating over the intervals between two collisions, when the distance to the nearest perturbing particle is large, simple asymptotic expressions may be used for the functions of relative motion. The collision instants themselves may in general be omitted from consideration.

Let us consider the collision of an atom with an electron. Before the collision, the motion of the atom is described by a plane wave \(e^{iKR}\). After the electron, having been scattered by the atom, has receded from it to a large distance, the motion of the atom will again be described

\(^*\) The problem of the collision of two particles reduces to the problem of the motion of a particle with the reduced mass in a central field. (Obviously, here one cannot restrict oneself to the one-dimensional case considered by Weisskopf.)

by a plane wave with the same value of the wave vector \(\mathbf K\), since the change of the atom’s momentum may be neglected because of the large difference in masses. But now this wave will have the form\({}^{13}\) \(e^{i\mathbf K\mathbf R+i2\eta_l}\), where \(\eta_l\) (the scattering phase) depends on the angular momentum of the collision \(\hbar\sqrt{l(l+1)}\) and the velocity of the relative motion \(v\). (The determination of \(\eta_l\) requires, generally speaking, the solution of the Schrödinger equation for the relative motion of the atom and the electron\({}^{60}\).) Thus, in the approximation of impact theory, the motion of an atom in an electron gas*) can be described by the function

\[ e^{i\mathbf K\mathbf R+i\sum^{t}2\eta_l}. \]

Here \(\sum^{t}2\eta_l\) is the total phase shift caused by collisions up to the time \(t\).

Substituting these functions into (49) and repeating the Weisskopf reasoning (this is now permissible, since the regions where the quasiclassical approximation breaks down have been excluded from consideration), one easily obtains for \(I(\omega)d\omega\), instead of (50), the following expression:

\[ I(\omega)d\omega \sim \left|A_{n'n''}\int e^{i(\omega-\omega_0)t-i\sum^{t}2(\eta_l'-\eta_l'')}\,dt\right|^2 . \tag{51} \]

Here one prime corresponds to the initial state and two primes to the final state.

The calculation of \(I(\omega)d\omega\) in the present case is reduced to the calculation of the spectrum of the oscillator

\[ f(t)=e^{i\omega_0t+i\sum^{t}2(\eta_l'-\eta_l'')}. \]

A similar calculation was already carried out in Section I. In the present case it requires a certain modification. When averaging the second factor in (12), one must now put the phase \(2(\eta_l'-\eta_l'')\) instead of \(\eta(\rho)\), and replace the integration over \(\rho\) by summation over \(l\). The number of collisions with angular momentum \(\hbar\sqrt{l(l+1)}\) during the time \(dt\) is equal to\({}^{60}\) \(Nv\,dt\,\dfrac{\pi}{k^2}(2l+1)\), where \(k=\mu v/\hbar\). Therefore

\[ \{\overline{e^{-i\eta'}-1}\} = Nv\,dt\,\frac{\pi}{k^2}\sum_{l=0}^{\infty}(2l+1) \left\{e^{-i2(\eta_l'-\eta_l'')}-1\right\} = -Nv\,dt(\sigma_r+i\sigma_i) \]

\[ \sigma_r=\frac{\pi}{k^2}\sum_{l=0}^{\infty}(2l+1) \left\{1-\cos 2(\eta_l'-\eta_l'')\right\}, \tag{52} \]

\[ \sigma_i=\frac{\pi}{k^2}\sum_{l=0}^{\infty}(2l+1) \sin 2(\eta_l'-\eta_l''). \tag{53} \]

*) Obviously, in the approximation of impact theory a similar consideration can be carried out for perturbing particles of any type.

It is easy to show that in the quasiclassical approximation expressions (52) and (53) pass into (13) and (14).

For large values of \(l\), when the motion is quasiclassical,

\[ \hbar \sqrt{l(l+1)} \simeq \mu v \rho = \hbar k \rho,\qquad \rho \simeq \sqrt{l(l+1)}/k,\qquad \Delta \rho \simeq \frac{(2l+1)\Delta l}{k\sqrt{l(l+1)}} \]

and

\[ \sum_{\Delta l} \frac{\pi}{k^2}(2l+1) \simeq 2\pi \rho\, d\rho . \]

Replacing the summation in (52) and (53) by integration, we again obtain (13) and (14). Moreover,\(^{22}\) in the quasiclassical approximation for scattering by the field \(\hbar D_n/R^n\),

\[ \eta_l = - \frac{\mu D_n k^{(n-2)}}{2\hbar l^{(n-1)}} \cdot \frac{\Gamma\!\left(\frac{1}{2}\right)\Gamma\!\left(\frac{n-1}{2}\right)} {\Gamma\!\left(\frac{n}{2}\right)} . \tag{54} \]

After substituting \(\hbar k = \mu v\) and \(k\rho \simeq l\) (recall also that \(\Gamma(1/2)=\sqrt{\pi}\)), expression (54) becomes

\[ \eta_l = - \frac{D_n}{2v\rho^{\,n-1}} \cdot \sqrt{\pi}\, \frac{\Gamma\!\left(\frac{n-1}{2}\right)} {\Gamma\!\left(\frac{n}{2}\right)}, \]

and \(2(\eta_l' - \eta_l'')\) exactly corresponds to \(\eta(\rho)\) from (5), if one puts \(C_n = D_n' - D_n''\).

Thus, the impact distribution of intensity (16) remains valid also when the condition of quasiclassicality is not satisfied. In this case \(\sigma_r\) and \(\sigma_i\), according to (52) and (53), are determined by the same phases \(\eta_l\) as the scattering.*)

The connection of impact broadening of lines with the general theory of collisions (first noted in \(^{13}\)) can be used in the calculation or estimation of the phases \(\eta_l\). (A presentation of methods for approximate estimates of \(\eta_l\) may be found in \(^{60}\).)

*) The total effective scattering cross section is defined, as is known,\(^{22}\) by the relation

\[ \sigma = \frac{4\pi}{k^2}\sum_{l=0}^{\infty}(2l+1)\sin^2 \eta_l . \]

Expression (52) can be rewritten in the form

\[ \sigma_r = \frac{2\pi}{k^2}\sum_{l=0}^{\infty}(2l+1)\sin^2(\eta_l' - \eta_l'') . \]

The difference between \(\sigma\) and \(\sigma_r\) consists in the fact that \(\sigma\) contains the phases \(\eta_l\) themselves, while \(\sigma_r\) contains the difference \(\eta_l' - \eta_l''\). We note that in \(^{23}\) the optical cross section was erroneously assumed to be equal to the scattering cross section \(\sigma\). This error was noted in \(^{13}\).

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